Persistent Homology and Equivariance in Data Analysis: A Topological Introduction

Abstract: This new book is intended as a first elementary introduction to Topological Data Analysis for mathematics students seeking a rigorous account of the foundations of persistent homology, as well as for computer scientists interested in its theoretical underpinnings. The exposition is as self-contained as possible: all the required background is recalled when needed, and only a few standard results are cited without proof. One section of the book, devoted to monodromy in biparameter persistence (Section 4.4), requires more advanced knowledge of algebraic topology. Persistent homology can be introduced from different perspectives, reflecting the variety of mathematical languages that have shaped its development over the years. Some approaches emphasize the algebraic foundations of the theory, while others highlight its topological essence. In this book, we adopt the latter viewpoint - the one that historically marked the birth of the subject - because we believe it offers both conceptual clarity and pedagogical effectiveness, making it particularly suitable for undergraduate and early graduate students. This book differs from existing introductory texts in several respects. First, it adopts a functional viewpoint: rather than representing data as finite (pseudo-)metric spaces, it treats them as functions encoding the information to be analyzed. This interpretative framework allows data to be viewed as measurable objects and highlights the role of observers and their equivariances in the analysis process. Second, this perspective provides a natural bridge between Topological Data Analysis and machine learning through the theory of Group Equivariant Non-Expansive Operators (GENEOs), which offers a mathematically grounded framework for incorporating symmetries and invariances into learning systems.

• The paper introduces a functional viewpoint in TDA, treating data as functions on topological spaces to enable measurability, observer-dependence, and symmetry.
• It establishes rigorous stability theorems linking bottleneck distances with uniform norms, ensuring robustness of persistent homology invariants.
• The study integrates group-equivariant frameworks via GENEOs, forming a theoretical bridge to symmetry-aware machine learning architectures.

Topological Foundations and Functional Viewpoint in Persistent Homology

"Persistent Homology and Equivariance in Data Analysis: A Topological Introduction" (2606.21084) offers a rigorous and self-contained exposition of the mathematics underlying Topological Data Analysis (TDA), focusing specifically on persistent homology and its extensions in the direction of group equivariance. The book is distinct in its explicit adoption of what the authors call the functional viewpoint, regarding data as functions on topological spaces rather than as finite metric spaces or point clouds. This paradigm facilitates a shift towards measurability, observer-dependence, and symmetry—in turn allowing deep connections with equivariant learning and pseudo-distances induced by transformation groups.

Persistent Homology: Topological and Algebraic Structure

The text begins with an extensive technical recall of the necessary topology and algebra, including (pseudo-)metrics, smooth manifolds, chain complexes, and the foundational frameworks of simplicial and singular homology. These constructions are recalled with particular attention to their categorical and functorial properties, ensuring that later constructions in persistent homology are implemented with full mathematical control.

Persistent homology is introduced via the construction of filtrations of topological spaces induced by continuous real-valued (or vector-valued) functions, bridging Morse-theoretic ideas with computational topology. The persistent Betti numbers function (PBNF)—encoding the rank of the image of homology induced by inclusion maps between sublevel sets—forms the basis for all subsequent developments. The work provides precise control of the relationships between these invariants and the canonical persistence diagram representation, emphasizing the important equivalence between the two (the Representation Theorem), under minimal regularity assumptions.

Stability Theory and Metrics

A significant aspect of the book is the formulation and proof of stability theorems connecting bottleneck distances (a.k.a. matching distance) between persistence diagrams with the uniform norm of filtering functions. These stability results are extended without loss of topological generality due to the functional viewpoint, which allows the use of the full machinery of continuous functions on compact domains. Explicit attention to the properties of right-continuity and the finiteness of persistent features ensures that the stability results are provable and applicable beyond the finite-data context.

The extension to biparameter persistent homology is treated with novel geometric and algebraic tools, including the Extended Pareto Grid formalism, which offers a precise framework to localize and interpret the coordinates of persistent features in the two-parameter setting. The book makes rigorous distinctions between regular, singular, and annihilation events for cornerpoints in persistence diagrams, highlights the monodromy phenomenon, and connects these to fundamental group representations in parameter space. These are nontrivial extensions that require careful management of homotopy, critical sets (e.g., Jacobi and Pareto critical points), and the behavior of contour-arcs under group actions.

Equivariance, Natural Pseudo-Distances, and GENEOs

A central theoretical contribution is the systematic integration of group actions into the persistent homology framework. The authors formalize perception pairs—function spaces equipped with transformation groups acting as isometries—and define the natural pseudo-distance as an optimization over group orbits. This notion captures the invariance and equivariance properties expected in high-level machine learning and geometry-encoding architectures.

From this setting, the book develops the theory of Group Equivariant Non-Expansive Operators (GENEOs), which serve as nonexpansive, group-equivariant mappings between function spaces. GENEOs abstract structural aspects of convolutional and symmetry-aware architectures, and the authors show that the set of such operators is closed under convex combinations and is compact under mild assumptions. The text establishes strict inequalities bounding the matching distance between persistence diagrams—after application of GENEOs—by the natural pseudo-distance, and demonstrates that under suitable conditions this bound is tight. This connection rigorously grounds the use of topological summaries as proxies for intrinsic, group-invariant metrics between data objects.

Theoretical Implications and Future Directions

The book's technical framework enables a deeper synthesis between TDA and equivariant ML, allowing for algorithmic approaches to shape, signal, and geometric data that are both stable and sensitive to group symmetries. The explicit consideration of biparameter and multiparameter persistence, as well as the precise management of the associated algebraic and topological structures, anticipates the needs of data analysis in high-dimensional, symmetry-rich, and observer-dependent regimes.

The treatment of monodromy and contour pairing, together with the action of the fundamental group of the parameter space, opens the door for further connections with sheaf-theoretic and categorical approaches to multi-parameter invariants, and with advanced algebraic structures such as persistence stacks and modules.

From a practical perspective, the axiomatic treatment of GENEOs as a module for equivariant data transformations forms a formal bridge to state-of-the-art equivariant neural architectures, with immediate implications for the theoretical understanding of expressivity and stability in geometric DL. Furthermore, the establishment of tightness for pseudo-metrics induced by GENEOs provides a theoretical guarantee for the faithfulness of TDA-based representations in applications where symmetries must be rigorously respected.

Conclusion

This text systematically develops the topological, algebraic, and metric foundations of persistent homology, with a unique emphasis on functional representations, group actions, and equivariance. The integration of natural pseudo-distances and GENEOs into the TDA framework provides a rigorous setting for both the analysis and construction of symmetry-respecting data analytic pipelines. The monograph thus serves as both a reference for foundational research and a blueprint for future advances at the intersection of geometry, topology, and machine learning.